While in the middle of posting this question, I found the answer haha.
I first entered the following into a terminal:
$ ibus engine
I then got the list of engines to find what I needed to change it to (output cropped for brevity):
$ ibus list-engine
xkb:ee::est - Estonian
xkb:sk:qwerty:slo - Slovak (qwerty)
xkb:sk::slo - Slovak
xkb:ro::rum - Romanian
xkb:jp::jpn - Japanese
anthy - Anthy
I then selected the Anthy engine (which also has support for English input so I don't need to keep swapping engines):
$ ibus engine anthy
Although there was no output for that command, using the keyboard shortcuts built in (Ctrl + ,) to go to the next input method (e.g. hiragana, katakana, english etc).
EDIT: I also found the way to swap between the engines (US to anthy) using a keyboard shortcut. First, open the ibus settings:
Selecting the Input Method tab, ensure the 'Customise active input methods' checkbox is ticked. Then, using the scrolldown (shown with the text 'Select an input method' to find the Japanese Anthy input method. Then click the 'Add' button on the right hand side of the screen. This will add 'Japanese - Anthy' to the list of Input Methods in Ibus. Now, when you press Ctrl + Space, it will properly switch between the English input method engine and the Japanese Anthy input method engine. The commands to do so via the terminal still work, this just enables it to be done via a keyboard shortcut.